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%I #14 Sep 08 2022 08:46:10
%S 2,3,5,10,15,23,34,37,47,70,259,275,278,497,563
%N Numbers m such that 6^m - m is a semiprime.
%C From _Robert Israel_, Sep 06 2016: (Start)
%C Even n is in this sequence iff (6^n-n)/2 is prime.
%C 3*k is in this sequence iff (2*6^(3*k-1)-k is prime.
%C Also contains 275, 278 and 683.
%C The only other possible member less than 275 is 259. (End)
%C a(16) >= 617. - _Kevin P. Thompson_, Apr 29 2022
%H FactorDB, <a href="http://factordb.com/index.php?id=1100000003337567816">Status of 6^617-617</a>
%e 2 is in this sequence because 6^2-2 = 2*17 is semiprime.
%e 10 is in this sequence because 6^10-10 = 2*30233083 and these two factors are prime.
%p Res:= NULL:
%p for n from 1 to 100 do
%p F:= ifactors(6^n-n, easy)[2];
%p if add(t[2], t=F) >= 3 or (hastype(F, symbol) and add(t[2], t=F) >= 2)
%p then flag:= false
%p elif add(t[2], t=F) = 2 and not hastype(F, symbol) then flag:= true
%p else
%p flag:= evalb(numtheory:-bigomega(6^n-n)=2)
%p fi;
%p if flag then Res:= Res, n fi
%p od:
%p Res; # _Robert Israel_, Sep 06 2016
%t Select[Range[90], PrimeOmega[6^# - #]== 2&]
%o (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [m: m in [1..90] | IsSemiprime(s) where s is 6^m-m];
%Y Cf. similar sequences listed in A252656.
%K nonn,more
%O 1,1
%A _Vincenzo Librandi_, Dec 21 2014
%E a(11)-a(15) from _Kevin P. Thompson_, Apr 29 2022
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